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Ch8: STRATEGY FOR INTEGRATION integration is more challenging than differentiation. No hard and fast rules can be given as to which method applies in a given situation, but we give some advice on strategy that you may find useful. how to attack a given integral, you might try the following four-step strategy. Ch8: STRATEGY FOR INTEGRATION 4-step strategy 1 Simplify the Integrand if Possible 2 Look for an Obvious Substitution function and its derivative 3 Classify the Integrand According to Its Form Trig fns, rational fns, by parts, radicals, 4 Try Again 1)Try subsitution 2)Try parts 3)Manipulate integrand 4)Relate to previous Problems 5)Use several methods Ch8: STRATEGY FOR INTEGRATION 4-step strategy 1 Simplify the Integrand if Possible Ch8: STRATEGY FOR INTEGRATION 4-step strategy 2 Look for an Obvious Substitution function and its derivative sin xdx 2 sec xdx cos xdx csc 2 xdx sec x tan xdx csc x cot xdx e dx x dx x dx x 2 1 Ch8: STRATEGY FOR INTEGRATION 4-step strategy 3 Classify the integrand according to Its form Trig fns, rational fns, by parts, radicals, 8.2 4 8.4 8.1 8.3 Try Again 1)Try subsitution 2)Try parts 3)Manipulate integrand 4)Relate to previous Problems 5)Use several methods Ch8: STRATEGY FOR INTEGRATION Ch8: STRATEGY FOR INTEGRATION 3 1 Classify the integrand according to Its form Integrand contains: 2 ln x ln and its derivative by parts Integrand contains: tan 1 , sin 1 f and its derivative by parts 4 Integrand radicals: a2 x2 , x2 a2 3 8.3 by parts (many times) poly 5 sin Integrand = f ( x) g ( x) n (x Integrand contains: only trig x cos xdx f (sin x) cos xdx f (tan x) sec xdx sec x tan xdx m n 2 m 6 We know how to integrate all the way 4 3 x)e3 x dx Integrand = rational PartFrac f & f’ 8.2 7 Back to original 2-times by part original x e sin xdx e cos xdx x 8 Combination: sec 3 xdx Ch8: STRATEGY FOR INTEGRATION x( x 1)e dx x tan xdx x 1 5 sin x dx 8 x cos x dx 1 3 102 dx (4 x 2 )3 / 2 x5 2 x 2 1 dx sin 2 x 1 cos 4 x dx cos x 122 x 2 dx dx 9 sin 2 x (4 x 2 )3/ 2 x e sin 2 xdx x 22 dx ( x 1)( x 1) dx 6 3 x 1 x csc 2 x cot 2 xdx Partial fraction sin( 3x) cos(2 x)dx 111 (2 tan x) x 2 112 2 x dx sin( 2 x)dx dx x 2 4 cos x cos 3 x dx (sin( 2 x) 2 cos x)e sin x dx Subsit or combination Trig fns dx x 6 x 1 sin x csc 2 x 1 dx x 3 1 4x2 sin(ln 15dx 2x2 x 2 radicals by parts x 2 dx x 2 )dx Ch8: STRATEGY FOR INTEGRATION 122 102 Partial fraction x5 2 x 2 1 dx dx 3 x 1 111 112 Subsit or combination x2 ( x 1)( x 2 1) dx (sin( 2 x) 2 cos x)e 15dx x3 2 x 2 x 2 cos x 9 sin x 2 dx sin x radicals dx (4 x 2 )3 / 2 dx dx x 1 x x Trig fns sin 5 x cos8 x dx csc 6 sin 2 x 1 cos 4 x dx 3 2 x cot 2 xdx (2 tan x) dx 2 cos x cos 3 x dx sin( 3x) cos(2 x)dx sin x csc 2 x 1 dx dx x 4 2 by parts x( x 1)e dx x x tan xdx x 1 e x sin 2 xdx sin(ln x 2 )dx 2 x 2 dx (4 x 2 )3 / 2 sin( 2 x)dx x 2 dx 1 4x 2 dx x 6 x 1 Ch8: STRATEGY FOR INTEGRATION Partial fraction Subsit or combination Trig fns radicals by parts Ch8: STRATEGY FOR INTEGRATION Partial fraction Trig fns Subsit or combination by parts radicals Ch8: STRATEGY FOR INTEGRATION Partial fraction Trig fns Subsit or combination by parts (Substitution then combination) radicals Ch8: STRATEGY FOR INTEGRATION Partial fraction Trig fns Subsit or combination by parts (Substitution then combination) radicals Ch8: STRATEGY FOR INTEGRATION CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS? if f (x ) YES or NO Continuous. Will our strategy for integration enable us to find the integral of every continuous function? Anti-derivative YES or NO F (x ) e x2 exist? dx Ch8: STRATEGY FOR INTEGRATION elementary functions. polynomials, trigonometric rational functions inverse trigonometric power functions hyperbolic Exponential functions inverse hyperbolic logarithmic functions all functions that obtained from above by 5-operations , , , , CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS? Will our strategy for integration enable us to find the integral of every continuous function? e x2 dx NO YES Ch8: STRATEGY FOR INTEGRATION elementary functions. FACT: polynomials, trigonometric rational functions inverse trigonometric power functions hyperbolic Exponential functions inverse hyperbolic logarithmic functions all functions that obtained from above by 5-operations If g(x) elementary , , , , g’(x) elementary NO: If f(x) elementary CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS? x Will our strategy for integration enable us to find the integral of every continuous function? F ( x) f (t )dt a need not be an elementary Ch8: STRATEGY FOR INTEGRATION CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS? Will our strategy for integration enable us to find the integral of every continuous function? FACT: NO: If g(x) elementary If f(x) elementary g’(x) elementary x F ( x) f (t )dt a need not be an elementary f ( x) e x2 has an antiderivative x F ( x) e dt a t2 is not an elementary. This means that no matter how hard we try, we will never succeed in evaluating in terms of the functions we know. In fact, the majority of elementary functions don’t have elementary antiderivatives.